Latent Variable

Latent Variable in Health Measurement, Preferences and Patient Classes

In health economics a latent variable is something a model treats as real but cannot observe directly, such as a patient's underlying health-related quality of life behind a set of questionnaire answers, the utility a respondent attaches to a treatment option in a discrete choice experiment, or membership of a subgroup whose preferences differ from the rest of the sample. Its value is inferred through a measurement model linking it to observed indicators, so every latent score or class assignment carries uncertainty. Latent variables sit behind item banks, new preference-based measures, choice models and some mapping methods, and a misread latent score can become a wrong utility value in a cost-utility model. The EQ-5D shows how reflective and formative indicators differ. The main methods include factor analysis, latent class analysis and structural equation models, and each has its own page.

What makes a variable latent

Bollen's review sets out several formal definitions of a latent variable, including local independence, which the measurement models described here assume. Under local independence the observed indicators are associated only because they share the latent variable, so once it is held fixed they are independent of each other. A primer on latent class analysis in ISPOR's Value & Outcomes Spotlight stresses that this applies within a latent class, and that across the whole sample the indicators may still be correlated.

"Latent" describes observability in a given dataset and model, not whether the phenomenon is real. A variable that is simply missing, with no indicators linked to it, cannot be estimated as a latent variable, and a single questionnaire item is not the construct it is meant to reflect.

Four families of latent variable model

Latent variable models are classified by whether the latent variable, and its indicators, are continuous or categorical. The same latent class primer reproduces this classification from Collins and Lanza.

Item response theory (IRT), which links a continuous trait to the probability of each response to an item, belongs in the latent trait cell. When the latent quantity changes over time the models extend to a latent growth model, or to latent transition analysis and the hidden Markov model for movement between unobserved states.

Reflective and formative indicators in the EQ-5D

Indicators can relate to a construct in two directions. Gamst-Klaussen, Gudex and Olsen describe effect (reflective) indicators as manifestations of the construct, which change when it changes, and causal (formative) indicators as items that drive a change in the construct. They note that factor analysis usually depends on a set of homogeneous items and is often not appropriate when both kinds are present.

Testing the five EQ-5D dimensions on EQ-5D-5L data from 7,933 respondents in six countries, they found the best fit when self-care and usual activities were effect indicators and pain/discomfort, anxiety/depression and mobility were causal indicators, and argued that this causal pattern supports the need for preference weighting. This makes a single reflective factor model a questionable choice for the EQ-5D.

Where latent variables enter health economic analysis

Latent variables appear at several points between collecting patient data and producing a cost per QALY. Each use below treats the unobserved quantity differently.

Item banks and new preference-based measures

The Patient-Reported Outcomes Measurement Information System (PROMIS) calibrated 11 item banks on 21,133 people with an item response model, the graded response model, after testing unidimensionality, local dependence and differential item functioning. IRT scores come out in logits, typically from about minus 4 to plus 4, and PROMIS reports every measure as a T-score with a mean of 50 and a standard deviation of 10 relative to the general population. The latent scale has no natural origin or unit until one is chosen.

Young and colleagues set out a five-step process for deriving a health-state classification from an existing non-preference-based instrument, in which factor analysis establishes the dimensions and Rasch analysis helps select items and reduce item levels. Applied to an overactive bladder questionnaire, it produced a five-dimension classification amenable to valuation.

Utility and preference classes in discrete choice experiments

In random utility models the utility itself is latent. Train states that the researcher does not observe the decision maker's utility, which splits into a representative part based on observed attributes and an unobserved part, so a discrete choice experiment estimates preference weights from choices alone.

Preference heterogeneity adds a second latent variable. Train shows that when the distribution of preference coefficients is discrete, mixed logit becomes the latent class model, in which each segment has its own coefficients and a population share estimated alongside them. Zhou, Thayer and Bridges reviewed 99 stated-preference studies in health that used latent class analysis. Over 90% used latent class logit models, about 80% reported two or three classes, and only 30% reported using statistical tests to detect significant differences in preferences between classes.

Self-reported health and response scales

Ordered response models treat an answer such as "good" or "fair" as the band in which an underlying latent index falls, with thresholds marking the bands. Lindeboom and van Doorslaer proposed a test that separates cut-point shift from index shift, using the Health Utilities Index Mark 3 as a more objective measure. They found clear evidence of both for age and gender, but not for income, education or language; such reporting heterogeneity may invalidate group comparisons and health inequality measures.

Hernández Alava, Pudney and Wailoo related the EQ-5D-5L and EQ-5D-3L through a joint ordinal regression model of the ten responses, including a latent factor for common unobserved influences across the five dimensions, such as a person's general tendency to "look on the bright side" or "take a dim view".

Mapping between instruments

A mapping function predicts a preference-based score from another measure. Where both measures carry measurement error, Lu, Brazier and Ades built a common factor model of the scales and showed that, for mapping treatment effects, ordinary least squares underestimates, and geometric mean regression overestimates, mappings from disease-specific to generic quality-of-life measures; ordinary least squares can be adjusted using external test-retest reliability. The NICE manual (PMG36, section 4.3.9) treats mapping to EQ-5D as a departure from the reference case and expects the chosen function to be justified.

Worked example of assigning a DCE respondent to a preference class

A latent class model gives each respondent a probability of belonging to each class, not a certain label. It follows from Bayes' rule, which Train uses to derive the preferences of people making a given sequence of choices.

$$\Pr(c \mid \mathbf{y}_n) = \frac{s_c \, P(\mathbf{y}_n \mid c)}{\sum_{m=1}^{M} s_m \, P(\mathbf{y}_n \mid m)}$$

where $\Pr(c \mid \mathbf{y}_n)$ is the probability that respondent $n$ belongs to class $c$, $\mathbf{y}_n$ is that respondent's sequence of choices, $s_c$ is the estimated population share of class $c$, $P(\mathbf{y}_n \mid c)$ is the probability of the sequence under class $c$'s preference weights, and $M$ is the number of classes.

All figures are illustrative. A two-class model of preferences for a new treatment estimates an effectiveness-focused class with share $s_1 = 0.6$ and a burden-averse class with share $s_2 = 0.4$. One respondent completed three choice tasks. Under class 1 weights the three choices have probabilities 0.5, 0.4 and 0.3, so $P(\mathbf{y}_n \mid 1) = 0.5 \times 0.4 \times 0.3 = 0.06$; under class 2 weights they have 0.8, 0.7 and 0.6, so $P(\mathbf{y}_n \mid 2) = 0.8 \times 0.7 \times 0.6 = 0.336$.

$$\Pr(1 \mid \mathbf{y}_n) = \frac{0.6 \times 0.06}{0.6 \times 0.06 + 0.4 \times 0.336} = \frac{0.036}{0.1704} = 0.211$$

where the denominator adds the class 2 product, 0.4 × 0.336 = 0.1344, to the class 1 product in the numerator. Class 2 membership therefore has probability 0.1344 / 0.1704 = 0.789.

If class 1 would pay GBP 20 a month to avoid a daily injection and class 2 GBP 90, assigning the respondent to the most likely class gives GBP 90, while weighting by the class probabilities gives 0.211 × 20 = 4.22 plus 0.789 × 90 = 71.01, a total of about GBP 75. The respondent is probably burden-averse, but a one-in-five chance of the other class remains after three tasks. Reading the modal class as certain ignores that probability, and a subgroup economic evaluation built on modal labels carries the misclassification silently into its results.

Assumptions that carry the inference

A model can fit well and still rest on assumptions the data cannot fully check. Five recur in health economic work.

1. Local independence. Shared item wording, a common response style or a second construct breaks it. PROMIS tested for local dependence, and the EQ-5D linking model adds a common latent factor to absorb shared response tendencies.
2. A chosen scale. Bollen lists identification and indeterminacy among the properties common to latent variable models; a one-unit rise in a latent score means nothing until the scale convention is stated.
3. Comparable measurement across groups. Differential item functioning and cut-point shift both mean that the same latent level produces different answers in different groups, so part of a group difference may come from the response scale.
4. The direction of the indicators. A reflective model is misspecified where some items cause the construct.
5. The number of classes. Information criteria, class size and interpretability all feed the choice, and different reasonable choices can produce different subgroups.

Using latent estimates in economic evaluation

The NICE manual (PMG36, section 4.3.2) describes deriving a health state utility as two elements, measuring health-related quality of life in people in the state, and valuing it according to preferences relative to other states. A latent trait score such as a PROMIS T-score does only the first. It reaches a QALY calculation through valuation of a health-state classification or a justified mapping whose uncertainty is carried into the model.

Latent classes describe how preferences differ, but a class supports a decision only if its members can be identified when the treatment decision is made. Labels such as "burden-averse" are interpretations added after estimation, and scores and class memberships should enter downstream models with their uncertainty.

Common misreadings of latent variable results

Most errors come from forgetting that a latent estimate is model-based. Two are common.

1. A trait score read as a utility. A factor score or T-score is not anchored where full health is 1 and dead is 0, and rescaling it does not make it a preference weight.
2. Fit read as validity. Good fit shows the data are consistent with the assumed structure, not that the construct is the intended one; construct validity needs hypotheses tested against other evidence.

Sources

• Bollen KA. Latent variables in psychology and the social sciences. Annual Review of Psychology. 2002;53:605-634. https://doi.org/10.1146/annurev.psych.53.100901.135239
• Cella D, Riley W, Stone A, Rothrock N, Reeve B, Yount S, et al. The Patient-Reported Outcomes Measurement Information System (PROMIS) developed and tested its first wave of adult self-reported health outcome item banks: 2005-2008. Journal of Clinical Epidemiology. 2010;63(11):1179-1194. https://doi.org/10.1016/j.jclinepi.2010.04.011
• Gamst-Klaussen T, Gudex C, Olsen JA. Exploring the causal and effect nature of EQ-5D dimensions: an application of confirmatory tetrad analysis and confirmatory factor analysis. Health and Quality of Life Outcomes. 2018;16:153. https://doi.org/10.1186/s12955-018-0975-y
• Hernández Alava M, Pudney S, Wailoo A. Estimating the relationship between EQ-5D-5L and EQ-5D-3L: results from a UK population study. PharmacoEconomics. 2023;41(2):199-207. https://doi.org/10.1007/s40273-022-01218-7
• Law EH, Harrington R. A primer on latent class analysis. Value & Outcomes Spotlight. November/December 2016:18-19. https://www.ispor.org/docs/default-source/publications/value-outcomes-spotlight/november-december-2016/vos-primer-on-latent-class-analysis.pdf
• Lindeboom M, van Doorslaer E. Cut-point shift and index shift in self-reported health. Journal of Health Economics. 2004;23(6):1083-1099. https://doi.org/10.1016/j.jhealeco.2004.01.002
• Lu G, Brazier JE, Ades AE. Mapping from disease-specific to generic health-related quality-of-life scales: a common factor model. Value in Health. 2013;16(1):177-184. https://doi.org/10.1016/j.jval.2012.07.003
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, updated March 2026. Sections 4.3.2 and 4.3.9.
• Train KE. Discrete Choice Methods with Simulation. 2nd ed. Cambridge: Cambridge University Press; 2009. Chapters 2, 6 and 11. https://eml.berkeley.edu/books/choice2.html
• Young T, Yang Y, Brazier JE, Tsuchiya A, Coyne K. The first stage of developing preference-based measures: constructing a health-state classification using Rasch analysis. Quality of Life Research. 2009;18(2):253-265. https://doi.org/10.1007/s11136-008-9428-0
• Zhou M, Thayer WM, Bridges JFP. Using latent class analysis to model preference heterogeneity in health: a systematic review. PharmacoEconomics. 2018;36(2):175-187. https://doi.org/10.1007/s40273-017-0575-4